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We propose higher-order approximation formulae recovering the surfaceelevation from the pressure at the bed and the background shear flow forsmall-amplitude Stokes and solitary water waves. They offer improvements overthe pressure transfer function and the hydrostatic approximation. The formulaecompare reasonably well with asymptotic approximations of the exact relationbetween the pressure at the bed and the surface wave in the zero vorticitycase, but they incorporate the effects of vorticity through solutions of theRayleigh equation. Several examples are discussed.

On the recovery of traveling water waves with vorticity from the pressure at the bed